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\title{《基础复分析》第2章点集拓扑基础 - 部分习题解答}
\author{CGZ ET AL}

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% 《基础复分析》习题二

\begin{enumerate}

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\item[1.]  % 1 

设 $(X, d)$ 为度量空间。证明 $(X, \delta)$ 为有界的度量空间，其中
    $$
    \delta(x, y) = \frac{d(x, y)}{1 + d(x, y)}.
    $$

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{\color{red}解答：
\begin{enumerate}[label=(\arabic*)]
\item 验证度量空间的定义。首先，$\delta$ 是一个从$X$到非负实数集的映射。
\item 自反性：验证 $\delta(x,y)=0$ 当且仅当 $x=y$.
\item 对称性：验证 $\delta(x,y)=\delta(y,x)$. 
\item 三角不等式：验证 $\delta(x,y)+\delta(y,z)\ge \delta(x,z)$. 
\end{enumerate}

}

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\item[2.]  % 2 

设 $d_1(x, y)$ 与 $d_2(x, y)$ 为空间 $X$ 上的两个度量。
证明由它们定义的拓扑相容，当且仅当任给 $x \in X$ 以及 $\varepsilon > 0$, 存在 $\delta > 0$, 使得 $d_1(x, y) < \delta$ 蕴涵 $d_2(x, y) < \varepsilon$; 反之亦然。

Suppose that there are given two distance functions $d_1(x,y)$ and $d_2(x,y)$ on the same space $S$.
They are said to be equivalent if they determine the same open sets.

Show that $d_1$ and $d_2$ are equivalent if to every $\varepsilon > 0$ there exists a $\delta > 0$ such that $d_1(x,y) < \delta$ implies $d_2(x,y) < \varepsilon$, and vice versa. Verify that this condition is fulfilled in the preceding exercise.

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{\color{red}解答：
\begin{enumerate}[label=(\arabic*)]
\item 设两个拓扑是相容的，即这两个拓扑规定的开集族是一样的。
\item 对任意 $x\in X$, 任意$\varepsilon>0$, 子集 $\{y\mid d_2(x,y)<\varepsilon\}$ 是度量 $d_2$ 定义的拓扑的开集。
\item 根据条件，这个子集在 $d_1$ 定义的拓扑中也是开集。
\item 根据开集的定义，存在 $\delta>0$, 使得子集 $\{y\mid d_1(x,y)<\delta\} \subseteq \{y\mid d_2(x,y)<\varepsilon\}$. 
\item 此时 $d_1(x, y) < \delta$ 蕴涵 $d_2(x, y) < \varepsilon$. 
\item 另一方面，对任意 $x\in X$, 任意$\varepsilon>0$, 子集 $\{y\mid d_1(x,y)<\varepsilon\}$ 是度量 $d_1$ 定义的拓扑的开集。
\item 根据条件，这个子集在 $d_2$ 定义的拓扑中也是开集。
\item 根据开集的定义，存在 $\eta>0$, 使得子集 $\{y\mid d_2(x,y)<\eta\} \subseteq \{y\mid d_1(x,y)<\varepsilon\}$. 
\item 此时 $d_2(x, y) < \eta$ 蕴涵 $d_1(x, y) < \varepsilon$. 
\item 这样就从两个拓扑的相容证明了两个度量之间关系。
\item 反过来，设两个度量满足条件：(i)对任意 $x\in X$, 任意$\varepsilon>0$, 存在 $\delta>0$, 使得 $d_1(x,y)<\delta$ 蕴涵 $d_2(x, y) < \varepsilon$. 以及(ii)对任意 $x\in X$, 任意$\varepsilon>0$, 存在 $\eta>0$, 使得 $d_2(x,y)<\eta$ 蕴涵 $d_1(x, y) < \varepsilon$. 我们要证明这两个拓扑是相容的。
\end{enumerate}

}

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% \item  % 3 

% 证明复平面在欧氏度量下定义的拓扑与在球面度量下诱导的拓扑相容。
    
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% \item  % 4 

% 证明度量空间内任意子集的聚点组成一个闭集。
    
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\item[5.]  % 5 

令 
$$
A = \{(x, y) \in \mathbb{R}^2 : x = 0, |y| \leq 1\}
$$

$$
B = \{(x, y) \in \mathbb{R}^2 : x > 0, y = \sin(1/x)\}
$$

证明 $A \cup B$ 是连通的。
    
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{\color{red}解答：
\begin{enumerate}[label=(\arabic*)]
\item 设 $X=U\cup V$, 其中 $U,V$ 是 $X$ 的两个互不相交的开集。
\item 设 $B\cap U\neq \varnothing$, 因为 $B$ 是连通的，所以(?) $B\subseteq U$.  
\item 因为 $U$ 也是 $X$ 中的闭集，所以 $\overline{B}\subseteq U$.
\item 因为 $\overline{B}=B\cup A=X$, 所以 $X=U$. 
\item 所以 $X$ 是连通的。
\end{enumerate}

}

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% \item  % 6 

% 设 $E \subsetneq \mathbb{C}$ 非空。证明 $\partial E$ 非空。
    
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% \item  % 7 

% 证明可分度量空间中的一个集合是可数的，如果集合中的所有点都是孤立的。
    
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% \item  % 8 

% 设 $E_1 \supseteq E_2 \supseteq \cdots$ 是 Hausdorff 空间中的非空紧致集序列，证明它们的交集非空。并举例说明如果 $E_i$ 只是闭的，则结论不一定成立。
    
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\item[9.]  % 9 

设 $X$ 是由所有实数序列 $\{x_n\}$ 组成的集合，使得每个序列只有有限项不为零。令 
$$
d(\{x_n\}, \{y_n\}) = \max |x_n - y_n|
$$

证明 $(X, d)$ 为不完备的度量空间。
    
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{\color{red}解答：
\begin{enumerate}[label=(\arabic*)]
\item 若一个度量空间中的柯西序列都收敛到这个空间中的点，则称这是完备的度量空间。
\item 记 $x_n=(1,\frac{1}{2},\frac{1}{3},\cdots,\frac{1}{n},0,0,\cdots )$. 
\item 若 $n\le m$, 则 $d(x_n,x_m)=\frac{1}{n+1}$. 
\item 因此 $\{x_n,n=1,2,3,\cdots\}$ 是一个柯西序列。
\item 这个序列的极限为 $x=(1,\frac{1}{2},\frac{1}{3},\cdots,\frac{1}{n},\cdots )$. 
\item 因为 $X$ 中的每个元素都只有有限项不为零，所以元素 $x$ 不属于 $X$, 所以 $X$ 不是完备的。
\end{enumerate}

}

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% \item  % 10 

% 证明扩充复平面在球面度量下是紧致的。
    
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% \item  % 11 

% 设 $X$ 与 $Y$ 是一个完备度量空间中的紧致集。
% 证明存在 $x \in X$, $y \in Y$ 使得 $d(x, y)$ 达到最小值。
    
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% \item  % 12 

% 证明由开集定义的连续映射与由度量定义的连续映射相容。
    
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\item[13.]  % 13 

试构造一个将单位圆盘 $|z| < 1$ 映为整个平面的同胚。

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{\color{red}解答：
\begin{enumerate}[label=(\arabic*)]
\item 两个拓扑空间之间若有双射，其映射和逆映射都是连续的，则称这两个拓扑空间是同胚的。
\item 设 $D=\{z\in\mathbb{C}\mid |z|<1\}$. 
\item 考虑 $z=re^{i\theta}\mapsto \tan(\pi r/2)e^{i\theta}$.
\item 验证这个双射是同胚。
\item 也可以考虑 $r\mapsto \frac{r}{1-r}$. 或其它 $(0,1)\leftrightarrow (0,\infty)$ 的连续双射。 
\end{enumerate}

}

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\end{enumerate}

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